Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation

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Global well-posedness of the KP-I initial-value problem in the energy space (2007, background): “Due to [ 23 ], it is known however that the inequalities (1.10), (1.11) cannot hold for any choice of the spaces F 1 (T ) and N 1 (T ); this forces us to approach the problem in a less perturbative way.”
Local Solutions in Sobolev Spaces with Negative Indices for the “Good” Boussinesq Equation (2008, methodology): “The same question was studied by Molinet, Saut and Tzvetkov [14]-[15], for the Benjamin-Ono (BO) equation ut + Huxx + uux = 0 (6) and for the Kadomtsev-Petviashvili 1 (KP1) equation (ut + uux + uxxx)x − uyy = 0, (7) respectively.”
On the Cauchy problem for higher-order nonlinear dispersive equations (2007, methodology): “These results are inspired by those from Molinet, Saut and Tzvetkov for the KPI equation [11] and the Benjamin-Ono (and the ILW) equation [12], (see also Bourgain [2] and Tzvetkov [20] for the KdV equation).”
Sharp well-posedness results for the BBM equation (2008, unknown): “We follow closely the proof of Lemma 4 in [20].”